The Black-Scholes pricing formula in the quantum context

Area of Science:

• * Quantum Finance
• * Mathematical Finance
• * Stochastic Calculus

Background:

• * Financial markets exhibit extreme price irregularities, defying classical explanations.
• * Quantum mechanics principles, such as uncertainty and observation effects, offer potential insights.
• * The Black-Scholes-Merton theory, a cornerstone of option pricing, relies on specific mathematical assumptions.

Purpose of the Study:

• * To explore quantum effects as a natural explanation for financial market irregularities.
• * To generalize the Wiener process and Ito theory within a quantum framework.
• * To extend the Black-Scholes option pricing formula into the quantum realm.

Main Methods:

• * Generalization of the Wiener process to incorporate quantum principles.
• * Demonstration of Wiener process differentiability in a Hilbert space context.
• * Extension of stochastic integration theory (Ito theory) to the quantum domain.

Main Results:

• * Quantum effects, specifically non-simultaneous observability and observer interference, are linked to market price irregularities.
• * The Wiener process is shown to be differentiable in a quantum context.
• * A generalized Ito theory and a quantum extension of the Black-Scholes option pricing formula are derived.

Conclusions:

• * Quantum mechanics provides a coherent framework for understanding complex financial market behaviors.
• * The generalized mathematical framework supports new approaches to stochastic integration and option pricing.
• * This research opens avenues for quantum-informed financial modeling and analysis.